Lorentz dynamics on closed 3-manifolds
Annales Henri Lebesgue, Volume 3 (2020), pp. 407-471.

Metadata

Keywords Lorentzian geometry

Abstract

In this paper, we give a complete topological, as well as geometrical classification of closed 3-dimensional Lorentz manifolds admitting a noncompact isometry group.


References

[Amo79] Amores, Angel M. Vector fields of a finite type G-structure, J. Differ. Geom., Volume 14 (1979) no. 1, pp. 1-6 | DOI | MR | Zbl

[AS97] Adams, Scot; Stuck, Garett The isometry group of a compact Lorentz manifold. I, II, Invent. Math., Volume 129 (1997) no. 2, p. 239-261, 263–287 | DOI | MR | Zbl

[BCD + 08] Barbot, Thierry; Charette, Virginie; Drumm, Todd; Goldman, William M.; Melnick, Karin A primer on the (2+1) Einstein universe, Recent developments in pseudo-Riemannian geometry (ESI Lectures in Mathematics and Physics), European Mathematical Society, 2008, pp. 179-229 | DOI | Zbl

[BM16] Bavard, Christophe; Mounoud, Pierre Extension maximale et classification des tores lorentziens munis d’un champ de Killing (2016) (https://arxiv.org/abs/1510.01253v2) | Zbl

[Car89] Carrière, Yves Autour de la conjecture de L. Markus sur les variétés affines, Invent. Math., Volume 95 (1989) no. 3, pp. 615-628 | DOI | MR | Zbl

[DG91] D’Ambra, Giuseppina; Gromov, Mikhael L. Lectures on transformations groups: geometry and dynamics, Surveys in Differential Geometry Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, (Cambridge, MA, 1990), American Mathematical Society, 1991, pp. 19-111 | DOI

[DM15] Dumitrescu, Sorin; Melnick, Karin Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist, Geom. Dedicata, Volume 179 (2015), pp. 229-253 | DOI | MR | Zbl

[DZ10] Dumitrescu, Sorin; Zeghib, Abdelghani Géométries Lorentziennes de dimension 3: classification et complétude, Geom. Dedicata, Volume 149 (2010), pp. 243-273 | DOI | Zbl

[D’A88] D’Ambra, Giuseppina Sometry groups of Lorentz manifolds, Invent. Math., Volume 92 (1988) no. 3, pp. 555-565 | MR

[FG83] Fried, David; Goldman, William M. Three-dimensional affine crystallographic groups, Adv. Math., Volume 47 (1983) no. 1, pp. 1-49 | DOI | MR | Zbl

[Fra02] Frances, Charles Géométrie et dynamique lorentziennes conformes (2002) (Ph. D. Thesis)

[Fra18] Frances, Charles Variations on Gromov’s open-dense orbit theorem, Bull. Soc. Math. Fr., Volume 146 (2018) no. 4, pp. 713-744 | DOI | MR | Zbl

[FZ02] Fisher, David; Zimmer, Robert J. Geometric lattice actions, entropy and fundamental groups, Comment. Math. Helv., Volume 77 (2002) no. 2, pp. 326-338 | DOI | MR | Zbl

[GK84] Goldman, William M.; Kamishima, Yoshinobu The fundamental group of a compact flat Lorentz space form is virtually polycyclic, J. Differ. Geom., Volume 19 (1984) no. 1, pp. 233-240 | DOI | MR | Zbl

[God91] Godbillon, Claude Feuilletages. Études géométriques, Progress in Mathematics, Volume 98, Birkhäuser, 1991 | MR | Zbl

[Gro88] Gromov, Michael M. Rigid transformation groups, Géométrie Différentielle (Travaux en Cours) Volume 33, Hermann, 1988, pp. 65-141 | MR | Zbl

[Kli96] Klingler, Bruno Complétude des variétés lorentziennes à courbure constante, Math. Ann., Volume 306 (1996) no. 2, pp. 353-370 | DOI | MR | Zbl

[KN63] Kobayashi, Shoshichi; Nomizu, Katsumi Foundations of differential geometry I, Interscience Publishers, 1963

[KR85] Kulkarni, Ravi; Raymond, Franck 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differ. Geom., Volume 21 (1985) no. 2, pp. 231-268 | DOI | MR | Zbl

[Mel11] Melnick, Karin A Frobenius theorem for Cartan geometries, with applications, Enseign. Math., Volume 57 (2011) no. 1-2, pp. 57-89 | DOI | MR | Zbl

[MS39] Myers, Sumner Byron; Steenrod, Norman Earl The group of isometries of a Riemannian manifold, Ann. Math., Volume 40 (1939) no. 2, pp. 400-416 | DOI | MR

[Nom53] Nomizu, Katsumi On the group of affine transformations of an affinely connected manifold, Proc. Am. Math. Soc., Volume 4 (1953), pp. 816-823 | DOI | MR | Zbl

[Orl72] Orlik, Peter Seifert Manifolds, Lecture Notes in Mathematics, Volume 291, Springer, 1972 | MR | Zbl

[Péc16] Pécastaing, Vincent On two theorems about local automorphisms of geometric structures, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 175-208 | DOI | Numdam | MR | Zbl

[Sal99] Salein, François Variétés anti-de Sitter de dimension 3 (1999) (http://www.umpa.ens-lyon.fr/~zeghib/these.salein.pdf) (Ph. D. Thesis) | Zbl

[Sco83] Scott, Peter The geometries of 3-manifolds, Bull. Lond. Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[Sha97] Sharpe, Richard W. Differential Geometry: Cartan’s generalization of Klein’s Erlangen Program, Graduate Texts in Mathematics, Volume 166, Springer, 1997 | MR | Zbl

[Tho14] Tholozan, Nicolas Uniformisation des variétés pseudo-riemanniennes localement homogènes (2014) (Ph. D. Thesis)

[Zeg96] Zeghib, Abdelghani Killing fields in compact Lorentz 3-manifolds, J. Differ. Geom., Volume 43 (1996) no. 4, pp. 859-894 | DOI | MR | Zbl

[Zeg98] Zeghib, Abdelghani Sur les espaces-temps homogènes. [Homogeneous spacetimes], The Epstein birthday schrift (Geometry and Topology Monographs) Volume 1, Geometry and Topology Publications, 1998, pp. 551-576 | DOI | MR | Zbl

[Zeg99a] Zeghib, Abdelghani Geodesic foliations in Lorentz 3-manifolds, Comment. Math. Helv., Volume 74 (1999) no. 1, pp. 1-21 | DOI | MR | Zbl

[Zeg99b] Zeghib, Abdelghani Isometry groups and geodesic foliations of Lorentz manifolds. I. Foundations of Lorentz dynamics., Geom. Funct. Anal., Volume 9 (1999) no. 4, pp. 775-822 | DOI | MR

[Zeg99c] Zeghib, Abdelghani Isometry groups and geodesic foliations of Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry group, Geom. Funct. Anal., Volume 9 (1999) no. 4, pp. 823-854 | DOI | MR | Zbl

[Zim86] Zimmer, Robert J. On the automorphism group of a compact Lorentz manifold and other geometric manifolds, Invent. Math., Volume 83 (1986) no. 3, pp. 411-424 | DOI | MR | Zbl